Manuscript X-D@3 · X-D-v3.tex · section · b7774529

Kernel formula and the multiplied-equation variant

Identification of the sharp prescription › Kernel formula and the multiplied-equation variant · lines 1065–1127

X-D@3#eq:limit-kernel, X-D@3#eq:homogeneous-difference

Write

C^{ρ} [h] = - \partial_{μ} a^{μ ρ} [h], (C^{*} b)^{ρ σ} = - \partial^{(ρ} b^{σ)} + \frac{1}{2} η^{ρ σ} \partial_{μ} b^{μ} .

For a compact covector test $a$ supported in the open slab, set $b = E_{0}^{A} a$ . X-A's ramp kernel identity and dominated convergence give

\int_{M_{T}} a_{ρ} X_{1, 0}^{ρ} d^{4} x = \int_{M_{T}} h_{ρ σ} θ (t) (C^{*} b)^{ρ σ} d^{4} x .

The background kernel is therefore

q_{a, 0} = θ C^{*} (E_{0}^{A} a) .

It is smooth on either side, locally bounded, and compactly supported after restriction to $t \geq 0$ . It can have a jump, but contains no hypersurface delta.

The multiplied-equation prescription instead solves

□ X_{1, 0} = C [h], X_{1, 0} (0) = \partial_{t} X_{1, 0} (0) = 0.

If $H_{0} [u_{0}, u_{1}]$ denotes the homogeneous wave solution, then

X_{1, 0} - X_{1, 0} = - H_{0} [0, a_{Σ}^{0 \cdot}] .

Thus the prescriptions differ in general, although they can agree for special backgrounds with vanishing relevant trace.

With $n = d t$ , define

B^{ρ σ} (b) = - n^{(ρ} b^{σ)} + \frac{1}{2} η^{ρ σ} n_{μ} b^{μ} .

The multiplied-equation kernel is

C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .

This distinction identifies the candidate; it does not prove joint sharp microcausal admission or define a quantum metric product.

Smooth coordinate convergence through $t = 0$ generally fails, as anticipated in the topology sheet. The completed scalar tests nevertheless converge because their supports remain separated from that surface.

Displayed formulas in this passage

X-D@3#eq:limit-kernel · line 1077
$\int_{M_{T}} a_{ρ} X_{1, 0}^{ρ} d^{4} x = \int_{M_{T}} h_{ρ σ} θ (t) (C^{*} b)^{ρ σ} d^{4} x .$
X-D@3#eq:homogeneous-difference · line 1099
$X_{1, 0} - X_{1, 0} = - H_{0} [0, a_{Σ}^{0 \cdot}] .$
unlabeled · line 1068
$C^{ρ} [h] = - \partial_{μ} a^{μ ρ} [h], (C^{*} b)^{ρ σ} = - \partial^{(ρ} b^{σ)} + \frac{1}{2} η^{ρ σ} \partial_{μ} b^{μ} .$
unlabeled · line 1084
$q_{a, 0} = θ C^{*} (E_{0}^{A} a) .$
unlabeled · line 1092
$□ X_{1, 0} = C [h], X_{1, 0} (0) = \partial_{t} X_{1, 0} (0) = 0.$
unlabeled · line 1108
$B^{ρ σ} (b) = - n^{(ρ} b^{σ)} + \frac{1}{2} η^{ρ σ} n_{μ} b^{μ} .$
unlabeled · line 1114
$C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .$